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Percolation transitions in compressed SiO2 glasses


Amorphous–amorphous transformations under pressure are generally explained by changes in the local structure from low- to higher-fold coordinated polyhedra1,2,3,4. However, as the notion of scale invariance at the critical thresholds has not been addressed, it is still unclear whether these transformations behave similarly to true phase transitions in related crystals and liquids. Here we report ab initio-based calculations of compressed silica (SiO2) glasses, showing that the structural changes from low- to high-density amorphous structures occur through a sequence of percolation transitions. When the pressure is increased to 82 GPa, a series of long-range (‘infinite’) percolating clusters composed of corner- or edge-shared tetrahedra, pentahedra and eventually octahedra emerge at critical pressures and replace the previous ‘phase’ of lower-fold coordinated polyhedra and lower connectivity. This mechanism provides a natural explanation for the well-known mechanical anomaly around 3 GPa, as well as the structural irreversibility beyond 10 GPa, among other features. Some of the amorphous structures that have been discovered mimic those of coesite IV and V crystals reported recently5,6, highlighting the major role of SiO5 pentahedron-based polyamorphs in the densification process of vitreous silica. Our results demonstrate that percolation theory provides a robust framework to understand the nature and pathway of amorphous–amorphous transformations and open a new avenue to predict unravelled amorphous solid states and related liquid phases7,8.

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Fig. 1: v-SiO2 local structures and connectivities.
Fig. 2: Sequence of percolation transitions in v-SiO2.
Fig. 3: Fractal percolating clusters at threshold.

Data availability

Figures and corresponding datasets (agr format), as well as sample trajectories at selected pressures are available at Zenodo (

Code availability

The DFTB+ code is publicly available at Additional information may be found there. The percolation code is available freely for non-commercial research at Zenodo ( Other codes for structural characterization are available from B.H.


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We thank the BioNano-NMRI team (L2C, UM) for computer facilities. A.H. thanks the CNRS (France) for funding, L. C. Rincón for introducing him to the SCC-DFTB method and E. Anglaret and F. Piuzzi for support that enabled him to participate in the conception of this project. This work was granted access to the high-performance computing resources of CINES by GENCI (Grand Equipement National de Calcul Intensif) under allocation grants nos. A0060910788, A0080910788 and A0100910788. BH acknowledges support from the French National Research Agency program PIPOG ANR-17-CE30-0009.

Author information




All authors initiated the project. A.H. and B.H. performed the MD tight-binding (SCC-DFTB) calculations of the pressurized glasses, after S.I. prepared the initial glass at ambient pressure by classical MD simulations. A.H. performed the tight-binding calculations of the crystals and computed the percolation tools. All authors contributed to data analysis: atomic structure (A.H., B.H. and S.I.), electronic structure and percolation (A.H.) and inelastic structure factors (S.I.). A.H. and B.H. developed the main conclusions and wrote the paper. S.I. contributed to the final version of the paper.

Corresponding authors

Correspondence to A. Hasmy or B. Hehlen.

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The authors declare no competing interests.

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Extended data figures and tables

Extended Data Fig. 1 Electronic structure of compressed v-SiO2.

Total DOS for a SiO2 glass at ambient pressure, with its corresponding projected DOS (a) and compared to different SiO4 crystalline polymorphs (b). Fermi energy (c) and total DOS (d) when the pressure increases. The results are compared with those corresponding to different SiO2 crystalline polymorphs.

Extended Data Fig. 2 Ionic bonding of v-SiO2.

Mulliken atomic charges for Si (a) and O (b), and the average Mulliken ionicity of the Si-O bond (c) in v-SiO2 as a function of pressure. The results (circles) are compared with those corresponding to different SiO2 crystalline polymorphs. The error bars in (b) correspond to the standard deviation of the average of the charges of all O atoms. Similar relative errors were estimated for (a) and (c).

Extended Data Fig 3 v-SiO2 atomic structures.

(a) SX(q) of our densified vitreous silicas compared to X-ray data reproduced from Prescher et al.3 and (b) Evolution of the maximum of the first sharp diffraction peak FSDP. (c) Calculated SN(q) compared to neutron data (black lines) reproduced from Zeidler et al.11.

Extended Data Fig. 4 v-SiO2 interatomic distances and angles.

(a) Calculated Si-O, O-O, and Si-Si distances at maximum of the distribution in our densified vitreous silicas. Si-O bond length are compared to X-Ray (squares) and neutron (+) scattering data. Si-Si distances are compared to those in the crystalline polymorphs. For stishovite, the interval corresponds to pressures between 10 GPa and 30 GPa. (b) Si-O-Si bond angle distribution (BAD) and pressure dependence of the Si-O-Si BAD marked by the arrow. The average value has been calculated from 110o to 175o. (c) Pressure dependence of the O-Si-O and examples of bond angle distributions (BAD).

Extended Data Fig 5 Face-shared SiOn polyhedra.

Number of face-sharing per polyhedron unit for dominant SiOn-SiOm connectivities as a function of pressure.

Extended Data Fig. 6 Percolation transitions.

(a) Percolation probability, P, versus v-SiO2 density for the different 4-, 5- and 6-folded coordinated Si, and their combinations. (b) P versus the fractions of SiOn.

Extended Data Fig. 7 OSiZ structures.

(a) Coordination numbers Z and Z′ of SiOZ polyhedra and OSiZ′ structures, (b) fraction of OSin, and (c) percolation probability of (OSi2-OSi2), (OSi2-OSi3), and (OSi3-OSi3) clusters.

Extended Data Table 1 Vitreous silica versus crystalline silicas

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Hasmy, A., Ispas, S. & Hehlen, B. Percolation transitions in compressed SiO2 glasses. Nature 599, 62–66 (2021).

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